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Generalizing Murray's law: An optimization principle for fluidic networks of arbitrary shape and scale

### Abstract

Murray's law states that the volumetric flow rate is proportional to the cube of the radius in a cylindrical channel optimized to require the minimum work to drive and maintain the fluid. However, application of this principle to the biomimetic design of micro/nano fabricated networks requires optimization of channels with arbitrary cross-sectional shape (not just circular) and smaller than is valid for Murray's original assumptions. We present a generalized law for symmetric branching that (a) is valid for any cross-sectional shape, providing that the shape is constant through the network; (b) is valid for slip flow and plug flow occurring at very small scales; and (c) is valid for networks with a constant depth, which is often a requirement for lab-on-a-chip fabrication procedures. By considering limits of the generalized law, we show that the optimum daughter-parent area ratio Γ, for symmetric branching into N daughter channels of any constant cross-sectional shape, is for large-scale channels, and for channels with a characteristic length scale much smaller than the slip length. Our analytical results are verified by comparison with a numerical optimization of a two-level network model based on flow rate data obtained from a variety of sources, including Navier-Stokes slip calculations, kinetic theory data, and stochastic particle simulations.

© 2015 AIP Publishing LLC. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Received 17 August 2015
Accepted 26 October 2015
Published online 06 November 2015

Acknowledgments:
We would like to thank Professor David Emerson for useful discussions and Professor Nicolas Hadjiconstantinou for providing the original LVDSMC source code. This work is financially supported in the UK by EPSRC Programme Grant Nos. EP/I011927/1 and EP/K038664/1.

Article outline:

I. INTRODUCTION
II. ANALYTICAL SOLUTIONS
A. The continuum-flow limit
1. Constant-shape networks
2. Constant-depth networks
B. The plug-flow limit
1. Constant-shape networks
2. Constant-depth networks
C. A slip flow approximation
1. Constant-shape networks
2. Constant-depth networks
III. NUMERICAL VERIFICATION AND DISCUSSION
A. Different shapes and sizes
B. Rarefied gas flow
C. Constant-depth rectangles
IV. CONCLUSION

/content/aip/journal/jap/118/17/10.1063/1.4935288

17.

17. K. Tesch, Task Quarterly 14, 227 (2010).

30.

30.To find the optimal branching angle, a second optimization should be performed with the parent inlet and daughter outlets fixed, so the channel lengths and branching angle are functions of the branching point.

31.

31.In both of these cases, there is a non-linear relationship between the pressure gradient and the mass flow rate, however as at least one of these quantities must be constant in the optimization (see Eq. (8)), Eq. (16) is still valid.

32.

32. F. M. White, Viscous Fluid Flow ( McGraw-Hill, 1974).

34.

34. G. Karniadakis, A. Beskok, and N. Aluru, Microflows and Nanoflows: Fundamentals and Simulation ( Springer, 2005).

35.

35. A. T. Conlisk, Essentials of Micro- and Nanofluidics With Applications to the Biological and Chemical Sciences ( Cambridge University Press, 2012).

38.

38. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids ( Clarendon Press, 1987).

39.

39. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Engineering Science Series Vol. 42 ( Clarendon Press, 1994).

46.

46.Our numerical solver is an in-house code written in MATLAB^{®}, which uses matrix inversion to calculate the mass flow rates. All simulations use a 100 × 100 mesh, which has been shown (via a grid resolution study) to provide mass flow rate solutions to within 1% of the values obtained using a 200 × 200 and 300 × 300 mesh.

47.

47.For our LVDSMC simulations, periodic boundary conditions were used in the streamwise direction, 10 deviational particles were used per cell, and the cell size Δx was chosen to be λ/5. The time-step Δt was set to , where is the most probable thermal velocity.

53.

53. S. K. Mitra and S. Chakraborty, Microfluidics and Nanofluidics Handbook ( CRC Press, 2012).

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/content/aip/journal/jap/118/17/10.1063/1.4935288

2015-11-06

2016-09-27

### Abstract

Murray's law states that the volumetric flow rate is proportional to the cube of the radius in a cylindrical channel optimized to require the minimum work to drive and maintain the fluid. However, application of this principle to the biomimetic design of micro/nano fabricated networks requires optimization of channels with arbitrary cross-sectional shape (not just circular) and smaller than is valid for Murray's original assumptions. We present a generalized law for symmetric branching that (a) is valid for any cross-sectional shape, providing that the shape is constant through the network; (b) is valid for slip flow and plug flow occurring at very small scales; and (c) is valid for networks with a constant depth, which is often a requirement for lab-on-a-chip fabrication procedures. By considering limits of the generalized law, we show that the optimum daughter-parent area ratio Γ, for symmetric branching into N daughter channels of any constant cross-sectional shape, is for large-scale channels, and for channels with a characteristic length scale much smaller than the slip length. Our analytical results are verified by comparison with a numerical optimization of a two-level network model based on flow rate data obtained from a variety of sources, including Navier-Stokes slip calculations, kinetic theory data, and stochastic particle simulations.

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